Strong confinement limit for the nonlinear Schr\"odinger equation constrained on a curve

Abstract

This paper is devoted to the cubic nonlinear Schr\"odinger equation in a two dimensional waveguide with shrinking cross section of order $\epsilon$. For a Cauchy data living essentially on the first mode of the transverse Laplacian, we provide a tensorial approximation of the solution $\psi^\epsilon$ in the limit $\epsilon\to 0$, with an estimate of the approximation error, and derive a limiting nonlinear Schr\"odinger equation in dimension one. If the Cauchy data $\psi^{\epsilon}_0$ has a uniformly bounded energy, then it is a bounded sequence in $H^1$ and we show that the approximation is of order $\mathcal O(\sqrt{\epsilon})$. If we assume that $\psi^{\epsilon}_0$ is bounded in the graph norm of the Hamiltonian, then it is a bounded sequence in $H^2$ and we show that the approximation error is of order $\mathcal O(\epsilon)$.